Analytic Continuation

What is the point of analytic continuation? Surely if a function is not valid in some of the complex plane and you change it so it becomes valid, you've created a new function, telling you nothing about the original one.

What is the point of analytic continuation? Surely if a function is not valid in some of the complex plane and you change it so it becomes valid, you've created a new function, telling you nothing about the original one.

Uh? Take for example the series . This series converges

and defines an analytic function within the unit disk, but we can continue its definition as to the whole

What is the point of analytic continuation? Surely if a function is not valid in some of the complex plane and you change it so it becomes valid, you've created a new function, telling you nothing about the original one.

Have you ever seen a function that wasn't defined at a point and then redefined to make it continuous in Calc II?